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Let $S=\langle n_1,...,n_r \rangle$ be a commutative semigroup, and let $I_S \subset k[x_1,...,x_r]$ the associated ideal of $S$, defined as the kernel of the polinomial map $\varphi:k[x_1,...x_n] \rightarrow k[t]$, defined by $\varphi(x_i)=t^{n_i}$ for all $i = 1,2,...,r$. It is known that for $r=3$ the ideal $I_S$ is a complete intersection or is generated by the 2x2 minors of a 2x3 matrix like $$ \begin{pmatrix} {x_1}^{a} & {x_2}^{b} & {x_3}^{c}\\ {x_2}^{d} & {x_3}^{e} & {x_1}^{f}\\ \end{pmatrix} $$ and so $I_S$, if is not a complete intersection, is a determinantal ideal. So I would ask:

  1. There are some conditions to establish when $I_S$ is a determinantal ideal for $r\geq4$ ?
  2. Is it possible generalize the case $r=3$? And so it could appens that for $r\geq4$, $I_S$ is the ideal generated by the 2x2 minors of a (r-1) x r matrix (or in general a (r-1) x m matrix)?
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  • $\begingroup$ For a general $(r-1)\times r$ matrix $A$ with entries in $k[x_{1},\ldots ,x_{r}]$, the locus where $\operatorname{rk}A\leq 1 $ has codimension $(r-1)(r-2)$ in $\mathbb{A}^r$. Therefore it is indeed a curve for $r=3$, but it is empty for $r\geq 4$. So the answer to 2. is very probably no — unless you find a very particular matrix $A$. $\endgroup$ – abx Jan 31 at 15:32

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